Luca Incurvati

On Class Hierarchies

Introduction

On class hierarchies. This paper refines Maddy's theory of classes, resolving rampant indeterminacy and issues with identity. It introduces supervaluational schemes & a bilateral natural deduction system.

Abstract

In her seminal article ‘Proper Classes’, Penelope Maddy introduced a novel theory of classes validating the naïve comprehension rules. The theory is based on a step-by-step construction of the extension and anti-extension of the membership predicate, which mirrors Kripke’s construction of the extension and anti-extension of the truth predicate. Maddy’s theory has been criticized by Øystein Linnebo for its ‘rampant indeterminacy’ and for making identity among classes too fine-grained. In this paper, I present a theory of classes which, while building on Maddy’s theory, avoids its rampant indeterminacy and allows for identity among classes to be suitably coarsegrained. I begin by presenting a bilateral natural deduction system for Maddy’s theory, which improves on her axiomatization in several respects. I then go on to show how to avoid the rampant indeterminacy by using supervaluational schemes in the construction of the extension and anti-extension of the membership predicate and how to augment the proof theory with corresponding, motivated rules. It turns out that whilst a van Fraassen-style supervaluational scheme suffices to avoid the basic problem of rampant indeterminacy, a supervaluational scheme based on maximally consistent extensions is needed for a proper treatment of identity.

Review

The paper "On Class Hierarchies" engages with a central debate in the foundations of mathematics concerning the nature of classes and their paradoxical properties. It critically examines Penelope Maddy's influential 'Proper Classes' theory, which offered a compelling approach to validating naïve comprehension rules through a Kripkean step-by-step construction of the membership predicate. The author acknowledges well-known criticisms of Maddy's framework, particularly those raised by Øystein Linnebo regarding 'rampant indeterminacy' and an overly fine-grained conception of class identity. The primary aim of this paper is to develop a refined theory of classes that retains the spirit of Maddy's constructivist approach while successfully overcoming these significant limitations. To achieve its stated objectives, the paper outlines a sophisticated technical program. It begins by presenting a novel bilateral natural deduction system for Maddy's original theory, an improvement that promises greater clarity and perhaps greater deductive power over her initial axiomatization. The core of the proposed solution to Maddy's issues lies in the innovative application of supervaluational schemes to the construction of the membership predicate's extension and anti-extension. The author meticulously details how a van Fraassen-style supervaluation adequately resolves the basic problem of rampant indeterminacy. Crucially, it is further demonstrated that a more robust supervaluational scheme, predicated on maximally consistent extensions, is essential for a proper, suitably coarse-grained treatment of class identity, along with the necessary augmentations to the proof theory. This paper represents a significant contribution to the philosophy of set theory and the ongoing efforts to provide coherent foundations for mathematics. By carefully building upon Maddy's seminal work and directly confronting its most pressing criticisms with novel technical machinery, the author offers a promising path forward. The integration of advanced proof theory with sophisticated supervaluational semantics for handling indeterminacy and identity is particularly commendable. Should the proposed framework successfully deliver on its promise to provide a theory of classes that is both robust and intuitive, it will undoubtedly stimulate considerable discussion and open new avenues for research in this complex and fundamental domain.

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